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Minimal Knowledge and Negation as FailurePowerPoint Presentation

Minimal Knowledge and Negation as Failure

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Outlines

- Propositional MBNF
- Positive MKNF
- General MKNF

- Extended MBNF with First-order Quantification
- Description Logics of MKNF
- ICs

Propositional MKNF

- Built from propositional symbols (atoms) using standard propositional connectives and two modal operators B and not.
B: “knowledge operator”K

not : “assumption operator”A

- Positive: if a formula or a theory (set of formulas) does not contain the negation as failure operator not.

Propositional MKNF

- Define when a positive formula F is true in a structure (I,S):
- (I,S) is a model of positive theory T if:
- (i) the axioms of T are true in (I,S)
- (ii) there is no (I’,S ’) such that S’ is a proper superset of S and the axioms of T are true in (I ’,S ’)

- S is maximized, so the believed propositions are minimized.

Propositional MKNF

- General MKNF: truth will be defined by a triple (I,Sb,Sn)
- (I,S) is a model of positive theory T if:
- (i) the axioms of T are true in (I,S,S)
- (ii) there is no (I’,S’) such that S ’ is a proper superset of S and the axioms of T are true in (I,S’,S)

Propositional MKNF

- An example:
- It is true in (I,S’S) when:

Then a model must satisfy:

(i)

(ii)

Three cases:

- F is tautology M=(I,S), S is the set of all interpretations.
- F is not tautology but a logical consequence of G no model
- F is not a logical consequence of G M=(I,Mod(G))

- It is true in (I,S’S) when:

Quantification

- Names: object constants representing all elements of |I |
- (I,S) is a model of positive theory T if:
- (i) the axioms of T are true in (I,S,S)
- (ii) there is no (I’,S’) such that S ’ is a proper superset of S and the axioms of T are true in (I,S’,S)

Quantification

- An example:
- Which courses are taught?
- Which courses are taught by known individuals?

MKNF-DL

- Goal:
- represent non-first-order features of frame systems

MKNF-DL

- A set of interpretations M is a model of Σif:
- (i) the structure (M,M) satisfies Σ
- (ii) for each set of interpretations M’, if M’M, then (M’,M) does not satisfy Σ

MKNF-DL

- An ideal rational agent trying to decide which set of propositions to believe.
- Set of prior beliefs + set of rules new beliefs
- “logical closure”
- Deduced set of beliefs coincides with the assumed believe assumed set is justified candidate for the agent to believe in
- Two kinds of beliefs:
- Beliefs that the agent assumed (A operator)
- New beliefs that derived (K operator)

ICs

- Example 1
- IC: Each known employee must be known to be either male or female.
Σ = <T,A> = <{},{employee(bob)}>

- IC: Each known employee must be known to be either male or female.

ICs

- Example 1

ICs

- Example 2
- IC: Each known employee has known social security number, which is known to be valid
Σ = <T,A> = <{},{employee(bob)}>

- IC: Each known employee has known social security number, which is known to be valid

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